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Evaluate $\int \frac{x}{\left ( x-3 \right )\sqrt{x+1}}dx$

Option 1)
$-2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Option 2)
$-2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Option 3)
$2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Option 4)
$2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Answers (1)

best_answer

As we learnt

Case for special type of indefinite integration -

$\int x^{m}(a+bx^{n})^{p}dx$

When $P$ is an integer if $P> 0$ then apply expanded form

$P< 0$ then we put $x=t^{k}$

- wherein

Where $k$ is the common denominator of $m$ and $n$

Put x + 1 = t². We get

$I=\int \frac{\left ( t^{2}-1 \right )2tdt}{\left ( t^{2}-4 \right )t}=2\int \frac{t^{2}-1}{t^{2}-4 }dt$

$=2\int \left \{ 1+\frac{3}{t^{2}-4} \right \}dt=2t+\frac{3}{2}\ln \left | \frac{t-2}{t+2} \right |+c$

$=2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$ .

Option 1)

$-2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Option 2)

$-2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Option 3)

$2\sqrt{x+1}+\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

Option 4)

$2\sqrt{x+1}-\frac{3}{2}\ln \left | \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right |+c$

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